I think to write the equation of a plane, all you need is the equation of the normal to that plane (which you have) and any point on the plane (which you have). If I’m not mistaken, it doesn’t matter which of the points you use, because there are multiple equations that describe the same plane. Can that be right? I might have to go look through my notebook from last quarter and reply again, haha.

Okay yeah I just checked my notebook.

The equation for a plane is in the form a(x-x0)+b(y-y0)+c(z-z0)=0 where <a, b, c> is the normal to the plane and <x0, y0, z0> is any point on the plane.

I could be wrong, but I don’t think you got the point of intersection correct. Just because the vectors along which the lines run aren’t parallel doesn’t mean the lines intersect. They could be skew.

E.g. the lines defined by (0, 0, u) and (1, v, 0) are not parallel but they never intersect.

To see if the two lines intersect, you need to match off the x, y & z coordinates.

So if your lines are (x1, y1, z1) and (x2, y2, z2) (where these are parametric functions) you solve:

x1(u) = x2(v)
y1(u) = y2(v)

This gets you a u and v. Plug those into z1 and z2 (respectively) and see if you get the same z coordinates. If so, they intersect. If not, they don’t.

If they intersect, call the point (xi, yi, zi). (i for point of intersection).

Now take the cross product and you get a vector (A, B, C).
So, take:
H = Axi + Byi + Czi

Whatever you get for H, your plane is:

Ax + By + Cz = H